Ion-optical system for mass separation

ABSTRACT

An ion optical system for use with a magnetic prism with a gap of width g and having an ion beam which is arranged to emerge from an elongated object slit and whose maximum angular dispersion Delta Beta max is smaller than g/l&#39;&#39; where l&#39;&#39; is the distance between the object slit and the effective entry boundary of the magnetic field associated with the magnetic prism, the system comprising means for causing the ion beam to converge in the vertical plane, so as to focus axially substantially into a line located substantially at the effective entry boundary of the magnetic field associated with the magnetic prism.

United States Patent [191 Chavet ION-OPTICAL SYSTEM FOR MASS SEPARATION [75] Inventor: Itzhak Chavet, Rehovoth, Israel [73] Assignee: The State of Israel, Atomic Energy Commission, Tel Aviv, Israel [22] Filed: May 9, 1972 [21] Appl. No; 251,820

Related US. Application Data [63] Continuation of Ser. No. 79,828, Oct. 12, 1970,

abandoned.

[30] Foreign Application Priority Data Oct. 17, 1969 lsrael 33204 [52] US. Cl. 250/41.9 ME, 250/4l.9 SB [51] Int. Cl. H0lj 39/34 [58] Field of Search 250/4l.9 SB, 41.9 ME

[56] References Cited UNITED STATES PATENTS 2,975,277 3/l96l Von Ardenne 250/41.9

[ 1 Nov. 20, 1973 3,122,631 2/1964 Geerk et a1. 250/4l.9

Primary Examine'rWilliam F. Lindquist Att0rneyBertram Ottinger [57] ABSTRACT An ion optical system for use with a magnetic prism with a gap of width g and having an ion beam which is arranged to emerge from an elongated object slit and whose maximum angular dispersion AB is smaller than g/l' where l is the distance between the object slit and the effective entry boundary of the magnetic field associated with the magnetic prism, the system comprising means for causing the ion beam to converge in the vertical plane, so as to focus axially substantially into a line located substantially at the effective entry boundary of the magnetic field associated with the magnetic prism.

4 Claims, 9 Drawing Figures Ob ect p// Magnet/c PATENTEDHBYZO ma $774,026 SHEET 30F 3 Q QC A, g} 4% Attorneys 1 ION-OPTICAL SYSTEM FOR MASS SEPARATION This is a continuation, division, of application Ser. No. 79,828 filed Oct. 12, 1970 now abandoned.

This invention concerns an improved ion optical system for use with magnetic prisms such as, for example, the magnetic sectors used in mass separators.

With a good ion optical system, as in a light optical system, the achievement of a high sharpness of the image, i.e., a high degree of resolution (of ions of different masses) should be combined with a high intensity of the ion beam. In contrast with analytical mass spectrometers in which the ion resolving power is of primary importance, in the case of mass separators the consideration of high intensity is not less important, since it determines the ion yield of the apparatus.

The relatively high intensity of the ion beam which it is desired to obtain rules out the use of electrostatic ion lenses in mass separators and, moreover, requires that the accelerated ions emerge from the ion source through relatively long (up to cm) exit slits (or object slits), and that the ion beam will have a considerably wide aperture.

This requirement, however, of mass separators is in conflict with the requirement of optimal ion-resolving power, since a long exit slit and a large aperture angle of the ion beam adversely affect the image qualities of the ion optical system. The use of long object slits and wider beam apertures introduces considerable image errors, causing increased blurring or smearing of the image. As a consequence, the separating power of the system is reduced, i.e., an efficient separation of the ions is prevented. This effect is even the more serious where ions of isotopes of higher masses are concerned, since in the magnetic-sector type instruments the distance between the images of ions of successive masses decreases with increasing mass.

In order to understand more clearly the theoretical considerations which give rise to the invention, reference is made to FIGS. 1, 2a and 2b of the accompanying drawings of which:

FIG. 1 shows a schematic plan view in the median (x,y)-plane of the ion optical system of a magnetic sector type mass separator, showing the main geometrical parameters;

FIG. 2a is a schematic representation of the left-hand part of the view shown in FIG. 1, showing the radial aperture angle a of one trajectory of the ion beam originating at a point located at a distance u from the median plane;

FIG. 2b is a schematic side view, in the vertical (x,z)- plane of the view shown in FIG. 2a, showing the axial aperture angle B of one trajectory of the ion beam originating at a point located ata distance s from the vertical plane;

FIG. 3a is a schematic plan view of an ion optical system according to the invention, showing three ion trajectories, the intermediate one being the principal trajectory;

FIG. 3b is a schematic side view of the view shown in FIG. 3a, showing a beam of ions (defined by three tra jectories), which converge to a focus in the axial (2) direction, according to the invention, which focus coincides with the magnetic field boundary;

FIG. 4a is a schematic horizontal cross sectional view of the ion source according to one embodiment of the invention;

FIG. 4b is a vertical cross section of the ion source shown in FIG. 4a showing the curved electrodes and magnetic field employed for generating the axially converging ion beam;

FIG. 5a is a schematic side view of an ion optical system of an ion beam collimator of the magnetic sector type; and

FIG. 5b is a schematic top view of the ion beam collimator corresponding to FIG. 5a.

In a system using magnetic prisms, such as a magnetic-sector type mass separator, the direction of propagation of the ion beam is defined as the x-axis (the longitudinal direction), the direction of the magnetic lines of force, which is perpendicular to x, is defined as the z-axis (axial direction), the y or radial direction being the third orthogonal axis and the origin of the coordinate system is located in the centre of the virtual object. The x-y and x-z planes are referred to as the median and vertical planes, respectively. (See FIGS. 1 and 2).

With a magnetic sector type mass separator the aberration of the image Ay in the y-axis is most decisive in the determination of the separating power, since it causes the blurring of the image in the intervals between neighbouring images, which may cause their mutual overlapping. w

This aberration is due to ions having trajectories which deviate slightly from the principal trajectory. The deviating trajectories are defined by the following variables: u, as, Bwhich are all shown in FIGS. 2a and 2b of the accompanying drawings. a and B are the radial and axial aperture angles, respectively. u and s are the horizontal and vertical distances, respectively of the point of origin of the trajectory from the vertical and median planes, respectively.

The radial aberration Ay can be expressed as a sum of terms in various powers of u, As and B and combinations of powers of these variables. These terms can be either homogeneous, such as 0z (a second order term),

or nonhomoge neous, such as s 3 (a third order term). All terms have appropriate coefficients A, which are complicated functions of the major geometrical parameters of the system, and of other minor parameters re- .lated to the fringe field of the magnetic prism. The

major parameters, most of which are shown in FIG. 1 of the accompanying drawings are:

4) the angle of the magnetic field sector.

5, e the angles of incidence and exit respectively, of the beam at the entry and exit boundaries of the magnetic field.

l, l" distance of object slit and image, respectively, from the magnetic field entry and exit boundaries respectively.

r', r the radii of curvature of these boundaries at the points of intersection of the principal trajectory.

t, t" the third order coefficients defining the third order profile of these boundaries.

As used in this specification the terms entry boundary and exit boundary of a magnetic field are the effective field boundaries of the magnetic prism, which have been determined taking into account the magnetic fringe fields and as contrasted with the physical boundaries of the magnetic prisms. The effective boundaries can be calculated, e.'g., according to R. I-Ierzog, Z. Naturf, 10a, 887 (1955) The geometrical parameters determining lengths are hereinafter expressed in pure numbers as ratios of the radius of deflection R, which is shown in FIG. 1, while angles are expressed in radians.

The invention is not concerned with the variable u and since the width of the exit slit is very small compared to its length it is assumed that u in the considerations and calculations hereinafter. In other words, the projection of the exit slit in the median x-y plane can be considered as a point source. On this assumption those aberration terms containing powers of u are neglected, hereinafter.

In order to achieve optimal image qualities in a particular system, it is a desiderata to choose the major geometrical parameters so that the sum of the aberration terms to the highest possible order is minimalized.

In practice this may be done by formulating mathematical expressions for the coefficients A, of the aberration terms up to the required order, involving the geometrical parameters mentioned above. These expressions are then equated to zero and a set of equations for the required parameters is thus obtained.

Mathematical expressions for the aberration terms up to the second order inclusive, in which the magnetic fringe fields have been taken into account, have been formulated by Wollnik, Nucl. Instr. and Meth. 34, 213 1965 A mathematical formulation of the aberration terms up to and including the third order terms has only been given without including the effects of fringe fields by Ludwig, Z. Naturf. 22a, 553 (1967). But even if such an expression were available, which would take into consideration the fringe field effects, its complexity makes it doubtful whether actual solutions for the various parameters could be reached, affording the desired minimalization of the aberration.

It is an object of the present invention to provide a new and improved ion optical system which permits the formulation of definite mathematical expressions for the coefficients of the aberration terms up to and including the third order, in which account is taken of the fringe field effects, and the practical solution of these equations, to determine the optimal geometrical parameters of the system.

According to the present invention there is provided in an ion optical system for use with a magnetic prism with a gap of width g and having an ion beam which is arranged to emerge from an elongated object slit and whose maximum angular dispersion AB as hereinafter defined is smaller than g/l as hereinbefore defined, means for causing the ion beam to converge in the vertical plane, so as to focus axially substantially into a line located substantially at the entry boundary as hereinbefore defined of the magnetic field associated with the magnetic prism.

In most ion sources used with such optical systems, every point of the object slit emits a narrow bundle of rays each deviating by some valve AB from the mean direction B originating at this point.

As indicated above, the present invention relates only to ion optical systems wherein this angular dispersion is relatively small, namely wherein:

where g is the width of the gap of the magnetic prism and l is as hereinbefore defined. In systems having larger angular dispersions AB, the axial focussing of the ion beam according to the invention is in practice unfeasible.

The variable AB which will appear in certain terms of the expression for the aberration, by virtue of it being very small in the systems concerned, may be regarded as a deviation of second order.

In consequence of this axial focussing of the ion beam, according to the invention, at the entry boundary of the magnetic field, all ion trajectories, having different radial aperture angles a, cross the entry boundary substantially in the median plane, i.e., their z coordinate is 0 at the entry boundary. The contribution of the fringe field at the entry boundary to the aberration of the image is thus eliminated, and as a consequence, a large number of terms in the mathematical expressions for the image aberration are cancelled. The resulting simplified equations, wherein the terms up to the third order inclusive are made equal to O, can now be actually solved to afford real values for the geometrical parameters.

Moreover, by this axial focussing of the beam, the homogeneous second and third order aberration terms in a can be cancelled through the proper choice of the profile of the entry boundary as expressed by r and t, which parameters were not involved in the expressions for the other optical conditions and had no effect on the particular solutions obtained for the other parameter s. Moreover, in order to correct for unavoidable discrepancies of the system as compared with the deducted theoretical values, these aberrations can still further be minimalized empirically by changing the profile of the entry boundary of the magnet (by shimming), without thereby upsetting the other characteristics of the optics, which have previously been determined through computation.

It is of further advantage to design the ion optical systern according to the invention, so that the ion beam will enter the magnetic field perpendicularly to the ef fective entry boundary, so as toachieve a maximum coincidence of its linear axial focus with the effective entry boundary.

The convergence and focussing of the ion beam can be effected in various ways. Thus, for example, the ion source can be designed with appropriately curved source and extraction electrodes, their common centre of curvature coinciding with the desired focus of the beam. Preferably and additionally it is arranged that the axis of the collimating magnetic field in the source should be curved.

Alternatively, when a relatively short object slit is used in the system, it is sufficient that only the extraction electrode is curved (while the source electrode is planar and the axis of the collimating magnetic field is straight). In this case, however, the centre of curvature of the extraction electrode should not be located at the entry boundary of the magnetic prism, but will be intermediate to this boundary and ion source.

While specific reference is made throughout the specification to the use of the invention in a mass separator, the advantages of the new ion optical system according to the invention are such as to render it equally applicable in other forms of equipment using magnetic prisms, e. g., in magnetic sector type ion collimators.

The axial focussing of the ion beam, according to the invention and as represented in FIG. 3b can be expressed as a definite correlation between B and s, namely:

wherein Z is the distance between the exit slit and the effective boundary of the magnetic field (see FIG. 1).

FIG. 3b further shows the ion beam which is axially focussed, according to the invention, and a random point at the object slit emitting a narrow bundle of rays deviating from B by the value AB, which is very small and may be regarded as a deviation of second order.

As stated hereinbefore, best results are obtained by the axial focussing of the ion beam according to the invention, when the optical system is designed so that the ion beam enters the magnetic field with normal incidence, that is to say, the principal trajectory intersects the entry boundary of the magnetic field at a right angle. In this case 6' in FIG. 1 becomes 0.

The mathematical treatment of such an ion optical system according to the invention is based on the following conditions:

a. Axial focussing of the beam expressed by B s/l' b. Normal incidence, that is e 0.

To this is added the main focussing condition, namely the known equation for first order radial focussing:

Because of the simplification introduced by the above condition in the mathematical expressions, the aberration Ay up to and including the third order involves only the four terms in: s' a; sAB; a; a.

The coefficients of the first two terms can now be mathematically formulated as a result of the simplification introduced by the axial focussing of the ion beam according to the invention.

Moreover, owing to the axial focussing, the expressions for these two aberration coefficients do not involve the parameters r and t of the entry boundary. Therefore the two equations wherein these aberration coefficients are made equal to zero can now be solved simultaneously, independent of the two remaining expressions for a and a 1 The equation concerning the elemination of the first term, namely making X za= 0, constitutes the requirement for a constant curvature of the image, independent of the radial aperture angle a. It has the form:

K =1 2tg e"/cos eI where I is a characteristic of the fringe-field geometry at the exit boundary of the magnetic prism. as

defined by Wollnik in the publication cited above.

also

G, is the radial magnification given by:

G, l"sin( e")/cos e" cos and D is the mass 2 The second term in SM? is a third order term (since AB is considered as a second order deviation) and concerns the aberration due to the axial angular dispersion AB. The requirement for the elimination of this term is expressed by the equation:

The equations in (l) and (2) above, when solved simultaneously together with the main focussing condition (c) above, will provide a two dimensional plurality of theoretical solutions for the five geometrical parameters ,l', l", r" and e".

(3) the requirement for second order radial focussing, i.e., elimination of the aberration term in a, can now be easily satisfied by determining r' (the radius of curvature of the entry boundary) according to the Hintenberger equations Z. Naturf. 12a, 377 (1957), in which any desired set of the geometrical parameters as previously obtained from (1) and (2) can be inserted.

(4) The remaining aberration term in a", expressing the third order aperture aberration, relates to the requirement for a third order radial focussing. In an ion optical system according to the invention, where the ion beam is focussed axially, this term can be eliminated, without disturbing the other qualities of the optical system, by giving the entry boundary a third order profile, which can be done by computation. Alternatively or additionally, the third order aberrations due to the discrepancies of the system as compared to the theoretical values, can be corrected empirically by shimrning, i.e., adding small pieces of iron sheets to the entry faces of the magnet. There is no special advantage in giving the exit boundary a profile higher than second order and in calculating t, t" (of the exit boundary) can be taken as 0.

It was shown above that the equations in (l) and (2) together with the condition (c) provide a two dimensional plurality of solutions for the major geometrical parameters. In practice, however, several additional conditions, arising out of technical or economical considerations, will further limit the choice of alternative solutions. For example, it may be desired to give the system a semi-symmetrical design, i.e., 1' =1", or it may be desired to make the total path length of the ions as small as possible for a given mass dispersion value D, in order to reduce the ion scattering by collision with molecules of the residual gases in the vacuum system.

By way of an example, the following values were calculated for the relevant parameters by the equations in (I) (3) above, using the value 1 0.03, and adding the limitation of 1/r'= 0.319, l/r -0.l79 (As stated above, the lengths are expressed as relative to R).

In another embodiment of the invention, the principle of axial focussing of the ion beam is applied to the ion optical system of a magnetic prism ion collimator (see FIGS. a, 5b). The function of such an instrument is to provide a beam of ions, which is parallel both in the axial (z) and the radial (y) direction.

Conventional ion beam collimators have usually employed magnetic and/or electrostatic lenses, and therefore were restricted to the use of small circular emission holes at the ion source. The present invention permits the use, in such a system, of a long emission slit, increasing considerably the intensity of the parallel beam thus obtained.

The calculation of the main parameters for'an ion beam collimator is comparatively simple, when the ion beam is axially focussed at the effective entry boundary of the magnetic field, according to the invention. This results in the same conditions as in the case of the mass separator described above, namely:

Here again it is advantageous to arrange for normal incidence of the beam in respect to the entry boundary, as expressed by:

The requirement for a parallel ion beam, whose focus may be assumed at infinity, imposes the further condition:

in the x, y and x,z planes.

Considering first the axial geometry (in the x-z plane) as represented in FIG. 5a, appropriate values for the magnet parameters for parallelism in the x-z plane are calculated. For these calculations the focus point of the beam at the entry boundary is regarded as a virtual object, and the following axial condition is obtained:

1/ tge" l 22g e"/cos 5" -1 It is then easy to derive an appropriate value of I in order to obtain with the above parameters a beam parallel also in the x-y plane (FIG. 5b), by applying thq-.. .ua1- m of ra alv{s Wh assume the form:

The main parameters calulated by the method described above for an actual embodiment, using the value of 1 0.03 are (e' 0 required) 4: e" 4040; I 1.48.

Curvatures of the entry and exit boundaries can be easily calculated.

FIGS. 4a and 4b show schematic cross sectional views in the horizontal and vertical planes, respectively, of an ion source according to one embodiment of the invention. The ion source consists, essentially, of a box 11 made of an electrically conducting material, one wall 12 of which is cylindrically curved and provided, at its center, with an exit slit or object slit 13 consisting of a narrow rectangular opening, its longitudinal axis being parallel to the axis of curvature of this wall 12 and to the z axis of the system. A side wall of the box is provided with a small aperture 14, for the entry of the ionizing electron beam (or the arc), originating at the cathode 15 which is located outside the box 11 and near the aperture 14. Outside the box 11 there are also provided means (not shown) for creating a curved magnetic field extending inside the source, whose curved axis 16 (FIG. 4b) lies in the vertical x-z plane. The purpose of this curved magnetic field is to curve the ionizing electron beam, so as to coincide with the axis 16 of the field. In addition this magnetic field has the usual effect of collimating the electron beam and concentrating it by decreasing its cross section.

An extraction electrode 17 is located a small distance from the curved front wall 12 of box 1 1 which is provided with the slit 13. This electrode is provided with a slit 18 which corresponds in shape to the object slit l3 and it is cylindrically curved to correspond with the curvature of the front wall 12 of box 11. The three centers of curvature in the (x,z)-plane of the axis 16 of the magnetic field of the ion source, of the front wall 12 provided with the exit slit 13 and of the extraction electrode 17 coincide at the point F (FIG. 4b) located at the effective entry boundary of the magnetic sector, which is also the desired focus of the ion beam in the axial (z) direction.

A high potential difference is maintained between the extraction electrode 17 and the ion source 11, the extraction electrode 17 being negative in respect with the ion source 11. The order of magnitude of this acceleration potential, as used in mass separators is about 10 to 10 volts.

The ions of the isotopes to be separated are generated by electron impact along the narrow region of the arc, whose center coincides with the axis 16 of the magnetic field of the ion source. The ions emerge out of the box 11 through the object slit 13, whereupon they are immediately subjected to the accelerating voltage, are propagated towards the extraction electrode 17 and by their acquired momentum pass through the slit 18 of this electrode in the direction of the focal point F on the entry boundary of the magnetic sector field.

I claim:

1. An ion optical system used with a homogeneous magnetic prism having pole pieces spaced apart a distance g establishing a magnetic field in the Z direction of an orthogonal coordinate system having axes X, Y and Z comprising:

a. an ion source for producing ions;

b. means defining an emission slit whose length s in the Z direction is large in comparison to its width in the Y direction for establishing the emitting area of the source;

c. an extraction system for shaping ions emitted from the slit into a beam whose principal trajectory is in the X-direction and which has a radial divergence angle C in the radial X-T plane;

d. means for causing the ion beam to converge in the axial X-Z plane and to be axially focused a distance I from the slit into a line perpendicular to the principal trajectory of the beam and lying in the median radial X-Y plane, where l' is the distance between the slit and the effective entry boundary of the field of the magnetic prism;

. the ion source, cooperating with the extraction system to cause the maximum angular dispersion Afl in the axial X-Z plane to be much smaller than g/l' where AB is the deviation in the X-Z plane of the trajectories of ions emitted from any point on the area of the source from a line connecting such point to the focus;

f. the parameters of the optical system being such that:

1. the angle of incidence e of the beam at the entry boundary of the magnetic field is zero for causing the principal trajectory of the beam to be normal to the entry boundary;

2. the distances 1' and l", where l" is the distance from the image to the magnetic field exit boundary, the angle 4) of the magnetic field sector, and the angle of exit e" of the beam at the exit boundary of the magnetic field all having values which satisfy the equation for first order redial focussing:

3 l, l", 4) and e" and the radius of curvature r" of the exit boundary being such that the coefficient, X of the aberration term in 01, satisfies the following equation:

where:

I B is a characteristic of the fringe-field geometry at the exit boundary of the magnetic prism 4. l, l", d), e" and r" being such that the coefficient of the aberration term in sA/3 satisfies the following equation:

wherein X and D are defined above.

2. An ion optical system according to claim 1, wherein the parameters of the optical system defining the entry boundary profile, namely the radius of curvature r of the entry boundary are such that the coefficient of the a aberration term is zero.

3. An ion optical system according to claim 2, wherein t, the coefficient of the third order term defining the third order profile of the entry boundary, has a value such that the coefficient of the a aberration term is zero.

4. An ion optical system according to claim 1,

wherein the parameter I" has the value of infinity in a both the X-Y and the X-Z planes whereby a collimated ion beam exits from the prism. 

1. An ion optical system used with a homogeneous magnetic prism having pole pieces spaced apart a distance g establishing a magnetic field in the Z direction of an orthogonal coordinate system having axes X, Y and Z comprising: a. an ion source for producing ions; b. means defining an emission slit whose length s in the Z direction is large in comparison to its width in the Y direction for establishing the emitting area of the source; c. an extraction system for shaping ions emitted from the slit into a beam whose principal trajectory is in the X-direction and which has a radial divergence angle Alpha in the radial X-Y plane; d. means for causing the ion beam to converge in the axial X-Z plane and to be axially focused a distance l'' from the slit into a line perpendicular to the principal trajectory of the beam and lying in the median radial X-Y plane, where l'' is the distance between the slit and the effective entry boundary of the field of the magnetic prism; e. the ion source, cooperating with the extraction system to cause the maximum angular dispersion Delta Beta max in the axial X-Z plane to be much smaller than g/l'' where Delta Beta is the deviation in the X-Z plane of the trajectories of ions emitted from any point on the area of the source from a line connecting such point to the focus; f. the parameters of the optical system being such that:
 1. the angle of incidence Epsilon '' of the beam at the entry boundary of the magnetic field is zero for causing the principal trajectory of the beam to be normal to the entry boundary;
 2. the distances l'' and l'''', where l'''' is the distance from the image to the magnetic field exit boundary, the angle phi of the magnetic field sector, and the angle of exit Epsilon '''' of the beam at the exit boundary of the magnetic field all having values which satisfy the equation for first order redial focussing: 1/l'''' tg phi - 1/l''/1 + tg phi /l'' - tg Epsilon ''''; 3 l'', l'''', phi and Epsilon '''' and the radius of curvature r'''' of the exit boundary being such that the coefficient, Xf, of the aberration term in s2 Alpha , satisfies the following equation: Xf Xd (Xc - Xbtg Epsilon '''' - 2tg Epsilon ''''/l''''cos2 Epsilon '''' - X1/2 phi 2 + 2D(1 - Gr)/l'''' phi 3) 2Xe (Xb + 1/l''''cos2 Epsilon '''' +D/l'''' phi 2) 0 where: Xi 1 + l''''/r''''cos3 Epsilon '''' (1 - cos phi ) + 2tg2 Epsilon ''''(1 - Cos phi ) + tg Epsilon ''''sin phi (2 - Gr) Grcos phi + sin phi (1 - Gr)/l''''Xe X1 + tg2 Epsilon '''' K'''' tg Epsilon ''''- tg Epsilon ''''/ phi + 1 - Gr/ phi 2 Xd 1/l'''' - tg Epsilon '''' + K'''' + 1/ phi Xc 3tg Epsilon ''''/r''''cos3 Epsilon '''' - 2Xntg2 Epsilon '''' + 4tg Epsilon ''''/ phi cos2 Epsilon '''' - tg2 Epsilon ''''/cos2 Epsilon '''' - 2tg2 Epsilon ''''/ phi 2 (1 - Gr) + 4K''''tg Epsilon ''''/cos2 Epsilon ''''Xb 1/r''''cos3 Epsilon '''' + tg3 Epsilon '''' - 2tg2 Epsilon ''''/ phi 2K''''tg2 Epsilon ''''Xa 1/cos2 Epsilon '''' + l''''/r''''cos3 Epsilon '''' - I '''' tg Epsilon ''''(5 + 6tg2 Epsilon '''')/cos Epsilon ''''K'''' 1 + 2tg2 Epsilon ''''/cos Epsilon '''' I ''''I '''' is a characteristic of the fringe-field geometry at the exit boundary of the magnetic prism Gr - l''''sin ( phi - Epsilon '''')/cos Epsilon '''' + cos phi D 1/2 ( (1-cos phi ) + l''''(sin phi + tg Epsilon ''''(1-cos phi ))); and
 4. l'', l'''', phi , Epsilon '''' and r'''' being such that the coefficient of the aberration term in s Delta Beta satisfies the following equation: 0 phi (l'' + phi )/l'' ( l''''/r''''cos3 Epsilon '''' + 1/cos2 Epsilon '''' - l''''(2 phi +l'')/ phi ( phi +l'') tg3 Epsilon '''' - l''''K''''tg2 Epsilon '''') + D/l'' wherein K'''' and D are defined above.
 2. An ion optical system according to claim 1, wherein the parameters of the optical system defining the entry boundary profile, namely the radius of curvature r'' of the entry boundary are such that the coefficient of the Alpha 2 aberration term is zero.
 2. the distances l'' and l'''', where l'''' is the distance from the image to the magnetic field exit boundary, the angle phi of the magnetic field sector, and the angle of exit epsilon '''' of the beam at the exit boundary of the magnetic field all having values which satisfy the equation for first order redial focussing: 1/l'''' tg phi - 1/l''/1 + tg phi /l'' - tg epsilon ''''; 3 l'', l'''', phi and epsilon '''' and the radius of curvature r'''' of the exit boundary being such that the coefficient, Xf, of the aberration term in s2 Alpha , satisfies the following equation: Xf Xd (Xc - Xbtg epsilon '''' - 2tg epsilon ''''/l''''cos2 epsilon '''' - X1/2 phi 2 + 2D(1 - Gr)/l'''' phi 3) -2Xe (Xb + 1/l''''cos2 epsilon '''' +D/l'''' phi 2) 0 where: Xi 1 + l''''/r''''cos3 epsilon '''' (1 - cos phi ) + 2tg2 epsilon ''''(1 - Cos phi ) + tg epsilon ''''sin phi (2 - Gr) -Grcos phi + sin phi (1 - Gr)/l''''Xe X1 + tg2 epsilon '''' -K'''' tg epsilon ''''- tg epsilon ''''/ phi + 1 - Gr/ phi 2 Xd 1/l'''' - tg epsilon '''' + K'''' + 1/ phi Xc 3tg epsilon ''''/r''''cos3 epsilon '''' - 2Xntg2 epsilon '''' + 4tg epsilon ''''/ phi cos2 epsilon '''' - tg2 epsilon ''''/cos2 epsilon '''' - 2tg2 epsilon ''''/ phi 2 (1 - Gr) + 4K''''tg epsilon ''''/cos2 epsilon ''''Xb 1/r''''cos3 epsilon '''' + tg3 epsilon '''' - 2tg2 epsilon ''''/ phi -2K''''tg2 epsilon ''''Xa 1/cos2 epsilon '''' + l''''/r''''cos3 epsilon '''' - I '''' tg epsilon ''''(5 + 6tg2 epsilon '''')/cos epsilon ''''K'''' 1 + 2tg2 epsilon ''''/cos epsilon '''' I ''''I '''' is a characteristic of the fringe-field geometry at the exit boundary of the magnetic prism Gr - l''''sin ( phi - epsilon '''')/cos epsilon '''' + cos phi D 1/2 ( (1-cos phi ) + l''''(sin phi + tg epsilon ''''(1-cos phi ))); and
 3. An ion optical system according to claim 2, wherein t'', the coefficient of the third order term defining the third order profile of the entry boundary, has a value such that the coefficient of the Alpha 3 aberration term is zero.
 4. An ion optical system according to claim 1, wherein the parameter l'''' has the value of infinity in both the X-Y and the X-Z planes whereby a collimated ion beam exits from the prism.
 4. l'', l'''', phi , epsilon '''' and r'''' being such that the coefficient of the aberration term in s Delta Beta satisfies the following equation: 0 phi (l'' + phi )/l'' ( l''''/r''''cos3 epsilon '''' + 1/cos2 epsilon '''' - l''''(2 phi +l'')/ phi ( phi +l'') tg3 epsilon '''' - l''''K''''tg2 epsilon '''') + D/l'' wherein K'''' and D are defined above. 